Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of 3−1+2⋊Q8 |
| automorphism | Aut(G) = C23×AGL(1,ℤ/9) (order 432) · Inn(G) = C2×AGL(1,ℤ/9) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #3663⋊C3 |
| Fitting | C8×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ M16 |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,5,6,7,2,8,3,9,4)(10,11,12,15,13,14,16,17);(2,4)(3,7)(5,6)(8,9)(11,14)(15,17)Polycyclic presentation:
2^2,3,2^2,3^2|b^2=d,d^2=e,f^3=g^2,b^a=be,f^a=f^2g,g^a=g^2,f^c=fg62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3663⋊C3, #63464⋊C3, M16×AGL(1,ℤ/9), … |
| maximal normal subgroup of | C2×#3663⋊C3, #63464⋊C3, M16×AGL(1,ℤ/9), … |
| maximal quotient of | (3−1+2⋊C4)⋊C8, #63375⋊C3, (C2×AGL(1,ℤ/9))⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 121);